Hmm... ok, so I think I figured it out. Even though it is uncharged, it has charged quarks. Unlike the electron, neutrons aren't elementary so I can't think of them in that way. This explains why it does precess, as my paper indicates you can actually observe the interfence pattern caused by a 2pi rotation (which introduces the new wavefunction to be the negative of the original).

Your formula (gq/2m)J is a classical oversimplification. For neutral (or charged) particles,
the magnetic moment is written as (mu)e/2m, where e is the magnitude of the electron charge, and e/2m (the nuclear magneton for hadrons) is just a convenient scale factor. In these units mu=+2.79 for the proton and -1.91 for the neutron.